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Chapter 21: Problem 48

You have a wire of length \(L=1.00 \mathrm{m}\) from which to make the square coil of a dc motor. The current in the coil is \(I=1.7 \mathrm{A},\) and the magnetic field of the motor has a magnitude of \(B=0.34\) T. Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

Short Answer

Expert verified
Single-turn torque is 0.036 Nm; two-turn torque is 0.018 Nm.

Step by step solution

01

Determine the Side Length of the Coil for Single-Turn

For a single-turn square coil, use the entire wire length to form the square. The perimeter of the square is equal to the length of the wire: \( L = 4s \). Thus, the side length \( s \) is given by: \[ s = \frac{L}{4} = \frac{1.00}{4} = 0.25 \text{ m} \].
02

Calculate the Area of the Coil for Single-Turn

The area \( A \) of a square is given by \( A = s^2 \). Substituting \( s = 0.25 \text{ m} \), we calculate the area: \[ A = (0.25)^2 = 0.0625 \text{ m}^2 \].
03

Calculate Maximum Torque for Single-Turn

The torque \( \tau \) on a coil is given by \( \tau = nIBA \sin \theta \), where \( n \) is the number of turns, \( I \) is the current, \( B \) is the magnetic field, and \( \theta \) is the angle between the field and coil normal. For maximum torque, \( \sin \theta = 1 \). Substituting \( n = 1 \), \( I = 1.7 \text{ A} \), \( B = 0.34 \text{ T} \), and \( A = 0.0625 \text{ m}^2 \):\[ \tau = 1 \times 1.7 \times 0.34 \times 0.0625 = 0.036125 \text{ Nm} \].
04

Determine Side Length for Two-Turn Coil

For a two-turn coil, the wire length shared by two squares gives \( 2L \) as the total perimeter \( 2L = 8s \). Solving for \( s \), \[ s = \frac{L}{8} = \frac{1.00}{8} = 0.125 \text{ m} \].
05

Calculate Area of the Two-Turn Coil

Using the same area formula for squares, the area \( A \) for a side \( s = 0.125 \text{ m} \) is: \[ A = (0.125)^2 = 0.015625 \text{ m}^2 \].
06

Calculate Maximum Torque for Two-Turn Coil

Using the same torque formula for \( n = 2 \) turns,\[ \tau = 2 \times 1.7 \times 0.34 \times 0.015625 = 0.0180625 \text{ Nm} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

dc motor
A DC motor is an essential electromechanical device that converts direct current (DC) electrical energy into mechanical motion. The core mechanism relies on the interaction between magnetism and electric current flowing through a coil. This coil is often part of a larger circuit, generating torque that turns the motor's rotor.

Here's how it works in simple terms:
  • When you pass current through a coil placed in a magnetic field, it experiences a force according to Lorenz's law. This is the foundation of the DC motor's operation.
  • By managing the direction of the current flow and using components like a commutator, DC motors produce continuous rotational motion, providing the torque needed to drive mechanical systems.
Understanding DC motors is fundamental when delving deeper into electric motor design, efficiency improvement, and robotics.
magnetic field
Magnetic fields are invisible forces that surround magnetic materials and electric currents, playing a critical role in the operation of motors. In our scenario with the DC motor, the magnetic field interacts with the current in the coil to produce motion. Let's explore some key points:
  • Magnetic fields are characterised by the magnetic flux density, commonly denoted as \( B \), and measured in teslas (T). In the problem, \( B = 0.34 \, \text{T} \).
  • This field exerts a force on the charged particles within the coil, generating motion depending on the arrangement and strength of both the field and the current.
  • The direction of the force is determined by the right-hand rule, helping us predict the behaviour of the coil within the magnetic field.
Thus, understanding magnetic fields is crucial for comprehending how electric and magnetic forces meld to drive mechanical systems.
current in coil
Current, denoted as \( I \), is the flow of electric charge through a conductor, such as the coil of wire in a motor. Here, the current in the coil is what allows the motor to do work by inducing a force in the magnetic field:
  • The problem specifies that the current is \( I = 1.7 \, \text{A} \), which is crucial for producing torque in a coil.
  • Electric current in the coil interacts with the magnetic field, and by using equations of electromagnetism, we can predict the resulting motion and torque.
  • Current direction matters: altering it changes the interaction dynamics, enabling the motor to maintain rotation in the desired direction.
A strong grasp of the role of current helps us harness electromagnetism to perform mechanical tasks efficiently.
square coil
A square coil is a loop of wire formed into a square shape, and it is pivotal in maximizing the exposure of wire to both magnetic fields and the forces generated within a DC motor.
  • In our exercise, a wire of length \( L = 1.00 \, \text{m} \) is used to form either a single-turn or a two-turn square coil.
  • The coil's effectiveness in generating torque relies heavily on its geometry. For a square coil: the perimeter \( L = 4s \), and each side length \( s \) can be calculated easily.
  • A single-loop coil utilizes the entire wire length to form a larger square, whereas a double-loop divides the wire, forming smaller squares. The side length directly affects the coil鈥檚 surface area, which in turn influences the torque generated.
Understanding how coil structure and turns apply to torque calculations helps optimize motor design and performance.

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Most popular questions from this chapter

A wire has a length of \(7.00 \times 10^{-2} \mathrm{m}\) and is used to make a circular coil of one turn. There is a current of \(4.30 \mathrm{A}\) in the wire. In the presence of a \(2.50-\mathrm{T}\) magnetic field, what is the maximum torque that this coil can experience?

An ionized helium atom has a mass of \(6.6 \times 10^{-27} \mathrm{kg}\) and a speed of \(4.4 \times 10^{5} \mathrm{m} / \mathrm{s} .\) It moves perpendicular to a \(0.75-\mathrm{T}\) magnetic field on a circular path that has a 0.012-m radius. Determine whether the charge of the ionized atom is \(+e\) or \(+2 e\).

Electron beams are sometimes used to melt and evaporate metals in order to deposit thin metallic films on surfaces (similar to gold plating). One method is to put the material to be evaporated (called the "target") into a small tungsten cup (a crucible that has a very high melting point) and direct a beam of electrons at the target. Your team has been given the task of designing an electron-beam evaporator. The crucible is a cylinder, \(2.0 \mathrm{cm}\) in diameter and \(1.5 \mathrm{cm}\) in height, and contains a small target of pure nickel (Ni). The electrons are accelerated through a potential difference of \(V=1.20 \mathrm{kV}\), and form a beam that originates below the crucible, exactly \(3.70 \mathrm{cm}\) off its center, in the \(+x\) direction (see the drawing). (a) What is the speed of the electrons in the beam? (b) You must steer the electron beam with a magnetic field so that it curls over the lip of the cup and strikes the nickel target. Assuming that a uniform field exists above the cup (the field is zero below), what must be the radius of the beam's circular path? (c) In what direction should the field point if the beam initially approaches the cup from the \(-y\) axis? (d) What must be the magnitude of the uniform magnetic field?

A copper rod of length \(0.85 \mathrm{m}\) is lying on a frictionless table (see the drawing). Each end of the rod is attached to a fixed wire by an unstretched spring that has a spring constant of \(k=75 \mathrm{N} / \mathrm{m} .\) A magnetic field with a strength of \(0.16 \mathrm{T}\) is oriented perpendicular to the surface of the table. (a) What must be the direction of the current in the copper rod that causes the springs to stretch? (b) If the current is 12 A, by how much does each spring stretch?

The magnetic field produced by the solenoid in a magnetic resonance imaging (MRI) system designed for measurements on whole human bodies has a field strength of \(7.0 \mathrm{T}\), and the current in the solenoid is \(2.0 \times 10^{2} \mathrm{A} .\) What is the number of turns per meter of length of the solenoid? Note that the solenoid used to produce the magnetic field in this type of system has a length that is not very long compared to its diameter. Because of this and other design considerations, your answer will be only an approximation.
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